Combinatorics. CS2100 Ross Whitaker University of Utah

Transcription

1 Combinatorics CS2100 Ross Whitaker University of Utah

2 Brain Teaser Three colleagues arrive at a hotel late, and there is one room left. They decide to share it. The hotel clerk says the room is $30 for the night, and each person puts $10 down on the counter. The travelers retire to their room. Meanwhile the hotel manager tells the clerk that room is only $25/night, and he gives the clerk 5, $1 bills, and tells him to refund the guests their $5. The clerk doesn t know how to split $5 three ways, so when he arrives at the room, he decides to give each guest one dollar, and keeps the remaining $2. Analysis: Each guest has paid $10-$1=$9, and thus the guests have paid $27 total. The clerk has $2 in his pocket $27 + $2 = $29!!!!! Where did the other dollar go?

3 Counting Outcomes of Events How many ways can two winners be chosen from four competitors (Andres, Barbara, Chyou, Dinesh)? How many elements are in the set: {{A,B}, {A,C}, {A,D}, {B,C}, {B,D}, {C,D}} What assumptions does this answer impose on the problem? Same person cannot win both prizes Order does not matter

4 Prizes Cont. What if there is a first prize and a second prize? What if these are two distinct door prizes (different awards)? What if these are two identical door prizes (same awards)? Are the prizes different? Can a person win both prizes? Yes No Yes No 12 6

5 Generally Does order macer? Are allowed? Yes No Yes Ordered list Unordered list No Set

6 Practice Dealing a five-card poker (draw) hand Every card is unique (no repetition) Order does not matter Dealing a two-card black-jack hand Order matters (one is down one is up) Repetition or not Creating game schedule for sports team Order matters Repetition or not Filling jar with various types of candy Order does not matter Repeats allowed

7 Organization in Counting How many permutations are there for the letters MATH? MATH, AMTH, AMHT, THAM, AHMT, HAMT, HMAT, MHAT, THMA, MHTA, HMTA, HATM, AHTM, MAHT, TMAH, MTHA, HTMA, TMHA, ATMH, TAHM, ATHM, TAMH, MTAH, HTAM How do we make sure we have them all?

8 Organizing and Counting Organize by letters: AMHT, MATH, HAMT, TAMH, AMTH, MAHT, HATM, TAHM, ATHM, MHTA, HMAT, THAM, AHTM, MHAT, HMTA, THMA, ATMH, MTAH, HTAM, TMAH, AHMT, MTHA, HTMA, TMHA Can organize within each column by second, third, etc.

9 Two Prizes for A,B,C,D Two different door prizes (order matters, repeats allowed) AA, AB, AC, AD BA, BB, BC, BD CA, CB, CC, CD DA, DB, DC, DD Two of the same door prize (order does not matter, repeats allowed) Idea: impose an order (alphabetical) AA, AB, AC, AD --, BB, BC, BD --, --, CC, CD --, --, --, DD

10 Combinatorial Equivalence One-to-one correspondence between finite sets Recognizing that two sets are the same size Example: How many multiples of 3 are there between 1 and 100 inclusive? 1*3=3, 2*3=6,, 3*33=99, 3*34=100 This is invertible {1,,33} {1, 3, 6, 99} How big is the set {1,,33}?

11 Combinatorial Equivalence Example: How many ways are there to distribute three coins (penny, nickel, dime) to 10 children? How many numbers are in the set {0, 1, 2, 3,, 998, 999}? Example: The number of binary sequences of length 10 The power set of {1, 2, 3,, 9, 10}.

12 Combinatorial Equivalence Example: Number of sets of size two from {1,, 9} Number of sets of size seven from {1,, 9} Example: The number of binary sequences of length 10 The power set of {1, 2, 3,, 9, 10}.

13 Combinatorial Equivalence Example How many positive-integer solutions to (x+y +z) = 21? How many two-element subsets of {1,,20} are there?

14 License plate More on Counting Either one or two letters from {A,L,B,M} followed by four or three numbers (respectively). Cases: One letter: arrange matrix/table with 4 columns and rows. Two letters: columns are {AA, AB, AL, AM, BA,, ML, MM} rows are =56000

15 Rule of Products If entries in list are created by i) selecting one of x objects and then ii) one of y objects, then the list has a total of x y entries. For finite sets A and B: AB = A B

16 More on Products How many numbers between 100 and 1000 have three distinct odd digits? Basic rules: three digits, {1,3,5,7,9}, and no repeats How many leaves are in this tree? 5 4 3=

17 Rule of Sums If a count/set can be split two disjoint pieces of size x and y, then the total size of the original list is x+y For disjoint sets A and B, AB = A + B

18 Rule of Sums Example: How many positive integers less than 1000 consist of distinct digits from the set {1,3,7,9}? Number of choices from Number of choices from =12 Number of choices from =24 Total = = 40

19 Rule of Sums Example: How many ways to win a dice game with three distinguishable dice, where a winning role contains a pair. Forms of winning roles: XXY, XYX, YXX, XXX (X and Y are distinct) Number X-Y choices are: 6 5=30 Three different configurations: =90 Number of solo X choices is 6 Total number of winning roles is 90+6=96

20 Rule of Sums Example: If we roll a die 3 times and record the results as an ordered list, how many ways are there to role exactly one 1. Divide into three sets with 1 in 3 different positions (first, second, third). Count number of choices for the other two rolls 3 (5 5)=75

21 Rule of Sums If a list to count can be split into two pieces of sizes x and y, and those pieces have x objects/cases in common, then the entire list has x+y-z entries Sets: AB = A + B - AB

22 Algorithms for Counting Odd numbers with distinct digits between 100 and 1000 for each U in {1,3,5,7,9} do for each H from 1 to 9 if H U for each T from 0 to 9 if T H and T U print H,T,U

23 Typical Mistake Number of ways to roll a sum of 10 on 3 distinct 6-side dice Choose an element of {1, 6} on first roll Choose an element of {1, 6} on second roll The third roll must be 10 minus the sum of first and second rolls

24 Example Number of ways to roll a sum of 10 on 3 distinct 6-side dice Choose an element of {1, 6} on first roll Choose an element of {1, 6} on second roll so that the sum is 9 or less The third roll must be 10 minus the sum of first and second rolls First two rolls: (number of ways to get >)+(number of ways to get 9) = (total number of cases for two rolls) Possibilities for first two rolls: 6 6 (3+2+1) = 30 Total number of possibilities is 30

25 General Formulas Total number of ordered lists from {1,,n} of length r is n r Number of permutations of length r on {1, n} is denoted P(n,r) P(n,n)=(n)(n-1) (3)(2)(1)=n! n factorial P(n,r)=(n!)/(n-r)!

26 P(n,r)=(n!)/(n-r)! Proof What if we assume P(n,n)=n! How can we prove P(n,n)=n!

27 Example Geography test List of 10 countries and 20 exports. Each country has a unique top export which is in the list (->10 exports in the list are bogus). How many possible guesses are there at the answer to this problem?

28 Counting with Equivalence Classes How many two-element subsets of {1,2,3,4} are there? Permutations vs subsets Two permutations are equivalent if they are different representations of the same subset Some equivalence classes: {{1,3}, {3,1}} {{4,2}, {2,4}}

29 Example How many ways to arrange 6 children in a circle. Consider 6 slots they must walk into: P(6,6) = 6! = 720 What if two arrangements are the same if every child is neighbors with the same two children? (rotate) How many ways are there to get the same arrangement in this new definition? How many distinct arrangements are there by this new definition?

30 Choosing Subsets C(n,r) is the number of subsets of {1,,n} of size r n choose r r-combinations from {1, n} C(n,r)=P(n,r)/r! Proof: Consider a single subset of length r. There are P(r,r)=r! distinct permutations of this set that are counted as part of P(n,r). Because every subset is counted r! times, the total number of r-combinations is P(n,r)/r!

31 r-combination Examples How many ways are there to draw a flush in 5-card poker (draw)?

32 Combinations A club of 10 women and 8 men is forming a committee of 5 people. How many different committees are possible? How many committees contain exactly 3 women How many committees contain at least 3 women Suppose Jack and Jill refuse to work together.

33 Binomials (1+x) n = (1+x) (1+x) [n times] Expand algebraically (1+x) (1+x) = (1 1)+(1 x)+(x 1)+(x x) = 1+2x+x 2 (1+x) 2 (1+x) = (1 1 1)+(1 x 1)+(x 1 1)+ (x x 1)+(1 1 x)+(1 x x)+(x 1 x)+(x x x) = 1+3x+3x 2 +x 3

34 Binomial Theorem The coefficient of the k th term of an expansion of (1+x) n is C(n,k) Examples: (1+x) 6 (1+x 2 ) 6

35 Pascal s Triangle

36 Binomial Expansions (1+x) n What if we substitute x=1? What does this say about the rows of Pascal s triangle? What does this say about the sum of k- combinations?

37 Binary Sequences Basic model for counting other things E.g. How many sequences are there with five 1 s and three 0 s? Idea: place the 1 s and then the 0 s Place the 1 s: C(8,5) Place the 0 s: C(3,3) C(8,5) C(3,3)=C(8,5)

38 Binary Sequences Theorem: the number of binary sequences with r 1 s and n-r 0 s is C(n,r)=C(n,n-r)

39 Example How many 10-letter sequences from the set {m,a,t} contain exactly 3 instances of the letter m? Idea: first place the m s into the ten slots C(10,3) How many ways are there to put {a,t} into the remaining 7 slots? 2 7 Answer: C(10,3) 2 7

40 Example From the previous answer, how many of those 10-letter strings contain exactly 4 a s? Algorithm: Place the m s C(10,3) Place the a s C(7,4) Place the t s C(3,3) Total: C(10,3) C(7,4)

41 Examples How many (distinguishable) arrangements of the letters in MISSISSIPPI? Does it matter what order our algorithm uses?

42 Unordered Lists (w/repitition) How many ways can we fill a bag with 10 pieces of 3 different kinds of fruit? How many solutions are there for nonnegative numbers in? x+y+z=10 Show these are the same question.

43 Unordered Lists 1. Bag containing r items chosen from n types 2. Binary sequence of length r+n-1 containing exactly r 0 s How are they the same? What is the solution to #2?

44 Bags (Unordered Lists) Correspondence. To each sequence A,B,C, associate the binary sequence consisting of A 0 s, 1, B 0 s, 1, C 0 s. Examples. A=3,B=3,C= A=0,B=9,C= A=0,B=0,C= A=4,B=3,C= et cetera

45 Theorem For natural numbers n and r 1. The number of solutions to the equation x1+ +xn=r using nonnegative integers is C(r+n-1,r). 2. The number of unordered lists of length r taken from a set of size n (w/repetition) is C(r+n-1,r) 3. The number of ways r items can be choosen from among n types is C(r+n-1,r)

46 Example How many different outcomes are possible from 4 tosses of a 6-sided die? How many of those outcomes sum to 14?

47 Summary on Data Types

48 Recursive Counting Reduce the counting problem to an easier problem and some modification to that problem. E.g. Find a recursive model for the number of games that will occur in the first round of a football tournament with n teams. Consider the tourney with n-1 teams. What changes in the schedule when the new team signs up?

49 Recursive Counting What is the recursive model for P(n,r), the number of r-permutations of {1,,n}? Choose an element of {1,,n} for the first position. Remove this element from the set to get {1,,n-1}. Fill in the rest of the r-1 positions from the n-1 set, which is P(n-1,r-1)

50 Permutations Continued Thus, P(n,r)=n P(n-1,r-1) What is the base case? P(n,0)=1 for all n

51 Example Let b n be the number of subsets of {1,,n} that do not contain consecutive numbers. E.g. for 4 we have {1,3}, {2,4}, {1,4}, {1}, {2}, {3}, {4}, {} Find a recursive model for b n Clearly all the elements in b n-1 can be in b n Additional elements would include the new element n, but we must exclude any subsets that contain n-1 This would be all of the elements in b n-2 b n = b n-1 + b n-2

52 Towers of Hanoi How to get discs from left peg to right Rules: One disc moves at a time to any peg No disc can reside above a smaller disc on the same peg

53 Towers of Hanoi Recursive solution For an n tower, the n-1 solution to the center peg would allow the largest disc to move all the way right. Then apply the n-1 solution to go from the center to right

54 Towers of Hanoi What is the recursive description of the number of moves needed to solve the the n sized TofH problem H n Move n-1 stack from left to middle is H n-1 Move n disc from left to right is 1 Move n-1 stack from middle to right is H n-1 Total is H n = 2H n-1 + 1

55 Towers of Hanoi Use induction to show that the number of moves for a n-sized Hanoi problem is H n = 2 n -1

56 Examples Find a recursive model for a n the number of n-digit numbers that do not contain the digit 0 Find a recursive model for a n, the number of ways to cover 2n grid of squares with 12 pieces